LMFDB - Embedded newform 2151.2.a.k.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$2.39589$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39589 q^{2} +3.74030 q^{4} +4.10699 q^{5} -3.62464 q^{7} +4.16958 q^{8} +O(q^{10})$	\(q+2.39589 q^{2} +3.74030 q^{4} +4.10699 q^{5} -3.62464 q^{7} +4.16958 q^{8} +9.83991 q^{10} +1.86664 q^{11} +3.35969 q^{13} -8.68425 q^{14} +2.50926 q^{16} +0.173143 q^{17} -1.09786 q^{19} +15.3614 q^{20} +4.47226 q^{22} -5.06374 q^{23} +11.8674 q^{25} +8.04946 q^{26} -13.5573 q^{28} +6.94337 q^{29} +6.11145 q^{31} -2.32723 q^{32} +0.414831 q^{34} -14.8864 q^{35} +3.44771 q^{37} -2.63035 q^{38} +17.1244 q^{40} -5.66600 q^{41} -5.01704 q^{43} +6.98178 q^{44} -12.1322 q^{46} -5.07918 q^{47} +6.13803 q^{49} +28.4330 q^{50} +12.5663 q^{52} -4.55248 q^{53} +7.66626 q^{55} -15.1132 q^{56} +16.6356 q^{58} -4.23002 q^{59} +8.14825 q^{61} +14.6424 q^{62} -10.5943 q^{64} +13.7982 q^{65} -8.96008 q^{67} +0.647606 q^{68} -35.6661 q^{70} +13.9037 q^{71} -6.34022 q^{73} +8.26034 q^{74} -4.10632 q^{76} -6.76589 q^{77} +1.38423 q^{79} +10.3055 q^{80} -13.5751 q^{82} -14.9390 q^{83} +0.711095 q^{85} -12.0203 q^{86} +7.78309 q^{88} +16.3643 q^{89} -12.1777 q^{91} -18.9399 q^{92} -12.1692 q^{94} -4.50888 q^{95} -17.5839 q^{97} +14.7061 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.39589	1.69415	0.847076	−	0.531472i	$-0.178361\pi$
$2$	2.39589	1.69415	0.847076	+	0.531472i	$0.178361\pi$
$3$	0	0
$3$	0	0
$4$	3.74030	1.87015
$5$	4.10699	1.83670	0.918351	−	0.395767i	$-0.129521\pi$
$5$	4.10699	1.83670	0.918351	+	0.395767i	$0.129521\pi$
$6$	0	0
$7$	−3.62464	−1.36999	−0.684993	−	0.728550i	$-0.740195\pi$
$7$	−3.62464	−1.36999	−0.684993	+	0.728550i	$0.740195\pi$
$8$	4.16958	1.47417
$9$	0	0
$10$	9.83991	3.11165
$11$	1.86664	0.562812	0.281406	−	0.959589i	$-0.409199\pi$
$11$	1.86664	0.562812	0.281406	+	0.959589i	$0.409199\pi$
$12$	0	0
$13$	3.35969	0.931811	0.465905	−	0.884835i	$-0.345729\pi$
$13$	3.35969	0.931811	0.465905	+	0.884835i	$0.345729\pi$
$14$	−8.68425	−2.32096
$15$	0	0
$16$	2.50926	0.627316
$17$	0.173143	0.0419933	0.0209966	−	0.999780i	$-0.493316\pi$
$17$	0.173143	0.0419933	0.0209966	+	0.999780i	$0.493316\pi$
$18$	0	0
$19$	−1.09786	−0.251865	−0.125933	−	0.992039i	$-0.540192\pi$
$19$	−1.09786	−0.251865	−0.125933	+	0.992039i	$0.540192\pi$
$20$	15.3614	3.43491
$21$	0	0
$22$	4.47226	0.953489
$23$	−5.06374	−1.05586	−0.527932	−	0.849287i	$-0.677032\pi$
$23$	−5.06374	−1.05586	−0.527932	+	0.849287i	$0.677032\pi$
$24$	0	0
$25$	11.8674	2.37347
$26$	8.04946	1.57863
$27$	0	0
$28$	−13.5573	−2.56208
$29$	6.94337	1.28935	0.644676	−	0.764456i	$-0.276992\pi$
$29$	6.94337	1.28935	0.644676	+	0.764456i	$0.276992\pi$
$30$	0	0
$31$	6.11145	1.09765	0.548824	−	0.835938i	$-0.315076\pi$
$31$	6.11145	1.09765	0.548824	+	0.835938i	$0.315076\pi$
$32$	−2.32723	−0.411401
$33$	0	0
$34$	0.414831	0.0711430
$35$	−14.8864	−2.51626
$36$	0	0
$37$	3.44771	0.566800	0.283400	−	0.959002i	$-0.408538\pi$
$37$	3.44771	0.566800	0.283400	+	0.959002i	$0.408538\pi$
$38$	−2.63035	−0.426698
$39$	0	0
$40$	17.1244	2.70761
$41$	−5.66600	−0.884880	−0.442440	−	0.896798i	$-0.645887\pi$
$41$	−5.66600	−0.884880	−0.442440	+	0.896798i	$0.645887\pi$
$42$	0	0
$43$	−5.01704	−0.765091	−0.382545	−	0.923937i	$-0.624952\pi$
$43$	−5.01704	−0.765091	−0.382545	+	0.923937i	$0.624952\pi$
$44$	6.98178	1.05254
$45$	0	0
$46$	−12.1322	−1.78879
$47$	−5.07918	−0.740875	−0.370437	−	0.928857i	$-0.620792\pi$
$47$	−5.07918	−0.740875	−0.370437	+	0.928857i	$0.620792\pi$
$48$	0	0
$49$	6.13803	0.876861
$50$	28.4330	4.02103
$51$	0	0
$52$	12.5663	1.74263
$53$	−4.55248	−0.625331	−0.312666	−	0.949863i	$-0.601222\pi$
$53$	−4.55248	−0.625331	−0.312666	+	0.949863i	$0.601222\pi$
$54$	0	0
$55$	7.66626	1.03372
$56$	−15.1132	−2.01959
$57$	0	0
$58$	16.6356	2.18436
$59$	−4.23002	−0.550701	−0.275351	−	0.961344i	$-0.588794\pi$
$59$	−4.23002	−0.550701	−0.275351	+	0.961344i	$0.588794\pi$
$60$	0	0
$61$	8.14825	1.04328	0.521639	−	0.853167i	$-0.325321\pi$
$61$	8.14825	1.04328	0.521639	+	0.853167i	$0.325321\pi$
$62$	14.6424	1.85958
$63$	0	0
$64$	−10.5943	−1.32429
$65$	13.7982	1.71146
$66$	0	0
$67$	−8.96008	−1.09465	−0.547324	−	0.836921i	$-0.684353\pi$
$67$	−8.96008	−1.09465	−0.547324	+	0.836921i	$0.684353\pi$
$68$	0.647606	0.0785338
$69$	0	0
$70$	−35.6661	−4.26292
$71$	13.9037	1.65006	0.825032	−	0.565086i	$-0.191157\pi$
$71$	13.9037	1.65006	0.825032	+	0.565086i	$0.191157\pi$
$72$	0	0
$73$	−6.34022	−0.742066	−0.371033	−	0.928620i	$-0.620996\pi$
$73$	−6.34022	−0.742066	−0.371033	+	0.928620i	$0.620996\pi$
$74$	8.26034	0.960245
$75$	0	0
$76$	−4.10632	−0.471027
$77$	−6.76589	−0.771044
$78$	0	0
$79$	1.38423	0.155738	0.0778688	−	0.996964i	$-0.475188\pi$
$79$	1.38423	0.155738	0.0778688	+	0.996964i	$0.475188\pi$
$80$	10.3055	1.15219
$81$	0	0
$82$	−13.5751	−1.49912
$83$	−14.9390	−1.63977	−0.819886	−	0.572527i	$-0.805963\pi$
$83$	−14.9390	−1.63977	−0.819886	+	0.572527i	$0.805963\pi$
$84$	0	0
$85$	0.711095	0.0771291
$86$	−12.0203	−1.29618
$87$	0	0
$88$	7.78309	0.829680
$89$	16.3643	1.73461	0.867307	−	0.497774i	$-0.165849\pi$
$89$	16.3643	1.73461	0.867307	+	0.497774i	$0.165849\pi$
$90$	0	0
$91$	−12.1777	−1.27657
$92$	−18.9399	−1.97463
$93$	0	0
$94$	−12.1692	−1.25515
$95$	−4.50888	−0.462602
$96$	0	0
$97$	−17.5839	−1.78537	−0.892687	−	0.450677i	$-0.851183\pi$
$97$	−17.5839	−1.78537	−0.892687	+	0.450677i	$0.851183\pi$
$98$	14.7061	1.48554
$99$	0	0
$100$	44.3876	4.43876
$101$	−9.02895	−0.898414	−0.449207	−	0.893428i	$-0.648293\pi$
$101$	−9.02895	−0.898414	−0.449207	+	0.893428i	$0.648293\pi$
$102$	0	0
$103$	2.28713	0.225358	0.112679	−	0.993631i	$-0.464057\pi$
$103$	2.28713	0.225358	0.112679	+	0.993631i	$0.464057\pi$
$104$	14.0085	1.37365
$105$	0	0
$106$	−10.9073	−1.05941
$107$	−9.34465	−0.903381	−0.451691	−	0.892175i	$-0.649179\pi$
$107$	−9.34465	−0.903381	−0.451691	+	0.892175i	$0.649179\pi$
$108$	0	0
$109$	−0.314568	−0.0301302	−0.0150651	−	0.999887i	$-0.504796\pi$
$109$	−0.314568	−0.0301302	−0.0150651	+	0.999887i	$0.504796\pi$
$110$	18.3675	1.75128
$111$	0	0
$112$	−9.09518	−0.859414
$113$	2.33072	0.219255	0.109628	−	0.993973i	$-0.465034\pi$
$113$	2.33072	0.219255	0.109628	+	0.993973i	$0.465034\pi$
$114$	0	0
$115$	−20.7967	−1.93931
$116$	25.9703	2.41128
$117$	0	0
$118$	−10.1347	−0.932972
$119$	−0.627580	−0.0575302
$120$	0	0
$121$	−7.51567	−0.683243
$122$	19.5223	1.76747
$123$	0	0
$124$	22.8587	2.05277
$125$	28.2042	2.52266
$126$	0	0
$127$	−0.790729	−0.0701658	−0.0350829	−	0.999384i	$-0.511170\pi$
$127$	−0.790729	−0.0701658	−0.0350829	+	0.999384i	$0.511170\pi$
$128$	−20.7284	−1.83215
$129$	0	0
$130$	33.0591	2.89947
$131$	−16.5237	−1.44368	−0.721842	−	0.692058i	$-0.756704\pi$
$131$	−16.5237	−1.44368	−0.721842	+	0.692058i	$0.756704\pi$
$132$	0	0
$133$	3.97933	0.345052
$134$	−21.4674	−1.85450
$135$	0	0
$136$	0.721932	0.0619052
$137$	−15.7754	−1.34778	−0.673891	−	0.738831i	$-0.735378\pi$
$137$	−15.7754	−1.34778	−0.673891	+	0.738831i	$0.735378\pi$
$138$	0	0
$139$	14.9098	1.26463	0.632317	−	0.774710i	$-0.282104\pi$
$139$	14.9098	1.26463	0.632317	+	0.774710i	$0.282104\pi$
$140$	−55.6795	−4.70578
$141$	0	0
$142$	33.3118	2.79546
$143$	6.27132	0.524434
$144$	0	0
$145$	28.5164	2.36815
$146$	−15.1905	−1.25717
$147$	0	0
$148$	12.8955	1.06000
$149$	16.5686	1.35735	0.678677	−	0.734437i	$-0.262554\pi$
$149$	16.5686	1.35735	0.678677	+	0.734437i	$0.262554\pi$
$150$	0	0
$151$	−11.4842	−0.934575	−0.467288	−	0.884105i	$-0.654769\pi$
$151$	−11.4842	−0.934575	−0.467288	+	0.884105i	$0.654769\pi$
$152$	−4.57760	−0.371292
$153$	0	0
$154$	−16.2103	−1.30627
$155$	25.0997	2.01605
$156$	0	0
$157$	−10.5567	−0.842520	−0.421260	−	0.906940i	$-0.638412\pi$
$157$	−10.5567	−0.842520	−0.421260	+	0.906940i	$0.638412\pi$
$158$	3.31646	0.263843
$159$	0	0
$160$	−9.55793	−0.755621
$161$	18.3543	1.44652
$162$	0	0
$163$	19.0184	1.48963	0.744816	−	0.667270i	$-0.232537\pi$
$163$	19.0184	1.48963	0.744816	+	0.667270i	$0.232537\pi$
$164$	−21.1925	−1.65486
$165$	0	0
$166$	−35.7923	−2.77802
$167$	1.53829	0.119036	0.0595181	−	0.998227i	$-0.481044\pi$
$167$	1.53829	0.119036	0.0595181	+	0.998227i	$0.481044\pi$
$168$	0	0
$169$	−1.71247	−0.131729
$170$	1.70371	0.130668
$171$	0	0
$172$	−18.7652	−1.43084
$173$	−16.0374	−1.21930	−0.609650	−	0.792671i	$-0.708690\pi$
$173$	−16.0374	−1.21930	−0.609650	+	0.792671i	$0.708690\pi$
$174$	0	0
$175$	−43.0150	−3.25163
$176$	4.68388	0.353061
$177$	0	0
$178$	39.2071	2.93870
$179$	8.92422	0.667027	0.333514	−	0.942745i	$-0.391766\pi$
$179$	8.92422	0.667027	0.333514	+	0.942745i	$0.391766\pi$
$180$	0	0
$181$	10.5224	0.782125	0.391062	−	0.920364i	$-0.372108\pi$
$181$	10.5224	0.782125	0.391062	+	0.920364i	$0.372108\pi$
$182$	−29.1764	−2.16270
$183$	0	0
$184$	−21.1137	−1.55652
$185$	14.1597	1.04104
$186$	0	0
$187$	0.323194	0.0236343
$188$	−18.9977	−1.38555
$189$	0	0
$190$	−10.8028	−0.783718
$191$	21.1875	1.53307	0.766537	−	0.642200i	$-0.221978\pi$
$191$	21.1875	1.53307	0.766537	+	0.642200i	$0.221978\pi$
$192$	0	0
$193$	−5.55065	−0.399545	−0.199772	−	0.979842i	$-0.564020\pi$
$193$	−5.55065	−0.399545	−0.199772	+	0.979842i	$0.564020\pi$
$194$	−42.1291	−3.02470
$195$	0	0
$196$	22.9581	1.63986
$197$	16.4993	1.17552	0.587761	−	0.809034i	$-0.300009\pi$
$197$	16.4993	1.17552	0.587761	+	0.809034i	$0.300009\pi$
$198$	0	0
$199$	20.6187	1.46162	0.730811	−	0.682580i	$-0.239142\pi$
$199$	20.6187	1.46162	0.730811	+	0.682580i	$0.239142\pi$
$200$	49.4820	3.49890
$201$	0	0
$202$	−21.6324	−1.52205
$203$	−25.1672	−1.76639
$204$	0	0
$205$	−23.2702	−1.62526
$206$	5.47973	0.381791
$207$	0	0
$208$	8.43035	0.584540
$209$	−2.04930	−0.141753
$210$	0	0
$211$	−28.3629	−1.95258	−0.976290	−	0.216466i	$-0.930547\pi$
$211$	−28.3629	−1.95258	−0.976290	+	0.216466i	$0.930547\pi$
$212$	−17.0277	−1.16946
$213$	0	0
$214$	−22.3888	−1.53047
$215$	−20.6049	−1.40524
$216$	0	0
$217$	−22.1518	−1.50376
$218$	−0.753672	−0.0510451
$219$	0	0
$220$	28.6741	1.93321
$221$	0.581706	0.0391298
$222$	0	0
$223$	2.14038	0.143331	0.0716653	−	0.997429i	$-0.477169\pi$
$223$	2.14038	0.143331	0.0716653	+	0.997429i	$0.477169\pi$
$224$	8.43539	0.563613
$225$	0	0
$226$	5.58415	0.371452
$227$	21.4911	1.42641	0.713207	−	0.700953i	$-0.247242\pi$
$227$	21.4911	1.42641	0.713207	+	0.700953i	$0.247242\pi$
$228$	0	0
$229$	−17.5694	−1.16102	−0.580511	−	0.814253i	$-0.697147\pi$
$229$	−17.5694	−1.16102	−0.580511	+	0.814253i	$0.697147\pi$
$230$	−49.8268	−3.28548
$231$	0	0
$232$	28.9510	1.90072
$233$	−12.7594	−0.835898	−0.417949	−	0.908470i	$-0.637251\pi$
$233$	−12.7594	−0.835898	−0.417949	+	0.908470i	$0.637251\pi$
$234$	0	0
$235$	−20.8601	−1.36077
$236$	−15.8215	−1.02990
$237$	0	0
$238$	−1.50361	−0.0974648
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−12.5216	−0.806589	−0.403294	−	0.915070i	$-0.632135\pi$
$241$	−12.5216	−0.806589	−0.403294	+	0.915070i	$0.632135\pi$
$242$	−18.0067	−1.15752
$243$	0	0
$244$	30.4769	1.95109
$245$	25.2088	1.61053
$246$	0	0
$247$	−3.68846	−0.234691
$248$	25.4822	1.61812
$249$	0	0
$250$	67.5743	4.27378
$251$	14.0327	0.885739	0.442869	−	0.896586i	$-0.353961\pi$
$251$	14.0327	0.885739	0.442869	+	0.896586i	$0.353961\pi$
$252$	0	0
$253$	−9.45216	−0.594252
$254$	−1.89450	−0.118872
$255$	0	0
$256$	−28.4744	−1.77965
$257$	5.67848	0.354214	0.177107	−	0.984192i	$-0.443326\pi$
$257$	5.67848	0.354214	0.177107	+	0.984192i	$0.443326\pi$
$258$	0	0
$259$	−12.4967	−0.776508
$260$	51.6095	3.20069
$261$	0	0
$262$	−39.5890	−2.44582
$263$	−5.16054	−0.318212	−0.159106	−	0.987261i	$-0.550861\pi$
$263$	−5.16054	−0.318212	−0.159106	+	0.987261i	$0.550861\pi$
$264$	0	0
$265$	−18.6970	−1.14855
$266$	9.53406	0.584571
$267$	0	0
$268$	−33.5134	−2.04716
$269$	1.69229	0.103181	0.0515903	−	0.998668i	$-0.483571\pi$
$269$	1.69229	0.103181	0.0515903	+	0.998668i	$0.483571\pi$
$270$	0	0
$271$	−11.7243	−0.712203	−0.356101	−	0.934447i	$-0.615894\pi$
$271$	−11.7243	−0.712203	−0.356101	+	0.934447i	$0.615894\pi$
$272$	0.434461	0.0263430
$273$	0	0
$274$	−37.7961	−2.28335
$275$	22.1521	1.33582
$276$	0	0
$277$	25.2439	1.51676	0.758380	−	0.651813i	$-0.225991\pi$
$277$	25.2439	1.51676	0.758380	+	0.651813i	$0.225991\pi$
$278$	35.7223	2.14248
$279$	0	0
$280$	−62.0699	−3.70939
$281$	23.7234	1.41522	0.707609	−	0.706604i	$-0.249774\pi$
$281$	23.7234	1.41522	0.707609	+	0.706604i	$0.249774\pi$
$282$	0	0
$283$	28.7226	1.70738	0.853690	−	0.520781i	$-0.174359\pi$
$283$	28.7226	1.70738	0.853690	+	0.520781i	$0.174359\pi$
$284$	52.0040	3.08587
$285$	0	0
$286$	15.0254	0.888471
$287$	20.5372	1.21227
$288$	0	0
$289$	−16.9700	−0.998237
$290$	68.3221	4.01201
$291$	0	0
$292$	−23.7143	−1.38778
$293$	−5.66344	−0.330862	−0.165431	−	0.986221i	$-0.552901\pi$
$293$	−5.66344	−0.330862	−0.165431	+	0.986221i	$0.552901\pi$
$294$	0	0
$295$	−17.3726	−1.01147
$296$	14.3755	0.835559
$297$	0	0
$298$	39.6966	2.29956
$299$	−17.0126	−0.983865
$300$	0	0
$301$	18.1850	1.04816
$302$	−27.5150	−1.58331
$303$	0	0
$304$	−2.75481	−0.157999
$305$	33.4648	1.91619
$306$	0	0
$307$	−17.7678	−1.01406	−0.507032	−	0.861927i	$-0.669257\pi$
$307$	−17.7678	−1.01406	−0.507032	+	0.861927i	$0.669257\pi$
$308$	−25.3065	−1.44197
$309$	0	0
$310$	60.1361	3.41550
$311$	24.8432	1.40873	0.704365	−	0.709838i	$-0.251232\pi$
$311$	24.8432	1.40873	0.704365	+	0.709838i	$0.251232\pi$
$312$	0	0
$313$	−29.3512	−1.65903	−0.829514	−	0.558485i	$-0.811383\pi$
$313$	−29.3512	−1.65903	−0.829514	+	0.558485i	$0.811383\pi$
$314$	−25.2928	−1.42736
$315$	0	0
$316$	5.17742	0.291253
$317$	−22.7860	−1.27979	−0.639894	−	0.768463i	$-0.721022\pi$
$317$	−22.7860	−1.27979	−0.639894	+	0.768463i	$0.721022\pi$
$318$	0	0
$319$	12.9607	0.725662
$320$	−43.5108	−2.43233
$321$	0	0
$322$	43.9748	2.45062
$323$	−0.190086	−0.0105766
$324$	0	0
$325$	39.8707	2.21163
$326$	45.5659	2.52366
$327$	0	0
$328$	−23.6248	−1.30446
$329$	18.4102	1.01499
$330$	0	0
$331$	23.2727	1.27918	0.639591	−	0.768715i	$-0.279104\pi$
$331$	23.2727	1.27918	0.639591	+	0.768715i	$0.279104\pi$
$332$	−55.8765	−3.06662
$333$	0	0
$334$	3.68557	0.201666
$335$	−36.7989	−2.01054
$336$	0	0
$337$	32.4686	1.76868	0.884338	−	0.466846i	$-0.154610\pi$
$337$	32.4686	1.76868	0.884338	+	0.466846i	$0.154610\pi$
$338$	−4.10290	−0.223169
$339$	0	0
$340$	2.65971	0.144243
$341$	11.4078	0.617769
$342$	0	0
$343$	3.12435	0.168699
$344$	−20.9189	−1.12787
$345$	0	0
$346$	−38.4238	−2.06568
$347$	−8.61985	−0.462738	−0.231369	−	0.972866i	$-0.574320\pi$
$347$	−8.61985	−0.462738	−0.231369	+	0.972866i	$0.574320\pi$
$348$	0	0
$349$	−20.5649	−1.10081	−0.550406	−	0.834897i	$-0.685527\pi$
$349$	−20.5649	−1.10081	−0.550406	+	0.834897i	$0.685527\pi$
$350$	−103.059	−5.50875
$351$	0	0
$352$	−4.34410	−0.231541
$353$	28.1526	1.49841	0.749206	−	0.662337i	$-0.230435\pi$
$353$	28.1526	1.49841	0.749206	+	0.662337i	$0.230435\pi$
$354$	0	0
$355$	57.1023	3.03068
$356$	61.2075	3.24399
$357$	0	0
$358$	21.3815	1.13005
$359$	−13.5612	−0.715735	−0.357867	−	0.933772i	$-0.616496\pi$
$359$	−13.5612	−0.715735	−0.357867	+	0.933772i	$0.616496\pi$
$360$	0	0
$361$	−17.7947	−0.936564
$362$	25.2106	1.32504
$363$	0	0
$364$	−45.5482	−2.38737
$365$	−26.0392	−1.36295
$366$	0	0
$367$	−25.7987	−1.34668	−0.673340	−	0.739333i	$-0.735141\pi$
$367$	−25.7987	−1.34668	−0.673340	+	0.739333i	$0.735141\pi$
$368$	−12.7063	−0.662360
$369$	0	0
$370$	33.9252	1.76368
$371$	16.5011	0.856695
$372$	0	0
$373$	35.5762	1.84207	0.921033	−	0.389484i	$-0.127347\pi$
$373$	35.5762	1.84207	0.921033	+	0.389484i	$0.127347\pi$
$374$	0.774339	0.0400401
$375$	0	0
$376$	−21.1781	−1.09217
$377$	23.3276	1.20143
$378$	0	0
$379$	−2.52878	−0.129894	−0.0649472	−	0.997889i	$-0.520688\pi$
$379$	−2.52878	−0.129894	−0.0649472	+	0.997889i	$0.520688\pi$
$380$	−16.8646	−0.865136
$381$	0	0
$382$	50.7630	2.59726
$383$	−13.7606	−0.703134	−0.351567	−	0.936163i	$-0.614351\pi$
$383$	−13.7606	−0.703134	−0.351567	+	0.936163i	$0.614351\pi$
$384$	0	0
$385$	−27.7874	−1.41618
$386$	−13.2988	−0.676890
$387$	0	0
$388$	−65.7691	−3.33892
$389$	−23.0695	−1.16967	−0.584836	−	0.811152i	$-0.698841\pi$
$389$	−23.0695	−1.16967	−0.584836	+	0.811152i	$0.698841\pi$
$390$	0	0
$391$	−0.876750	−0.0443391
$392$	25.5930	1.29264
$393$	0	0
$394$	39.5304	1.99151
$395$	5.68500	0.286043
$396$	0	0
$397$	16.1634	0.811218	0.405609	−	0.914047i	$-0.367059\pi$
$397$	16.1634	0.811218	0.405609	+	0.914047i	$0.367059\pi$
$398$	49.4003	2.47621
$399$	0	0
$400$	29.7784	1.48892
$401$	−13.9016	−0.694215	−0.347107	−	0.937825i	$-0.612836\pi$
$401$	−13.9016	−0.694215	−0.347107	+	0.937825i	$0.612836\pi$
$402$	0	0
$403$	20.5326	1.02280
$404$	−33.7710	−1.68017
$405$	0	0
$406$	−60.2980	−2.99254
$407$	6.43562	0.319002
$408$	0	0
$409$	−7.79717	−0.385545	−0.192773	−	0.981243i	$-0.561748\pi$
$409$	−7.79717	−0.385545	−0.192773	+	0.981243i	$0.561748\pi$
$410$	−55.7529	−2.75344
$411$	0	0
$412$	8.55457	0.421454
$413$	15.3323	0.754453
$414$	0	0
$415$	−61.3545	−3.01177
$416$	−7.81879	−0.383348
$417$	0	0
$418$	−4.90990	−0.240151
$419$	4.44671	0.217236	0.108618	−	0.994084i	$-0.465357\pi$
$419$	4.44671	0.217236	0.108618	+	0.994084i	$0.465357\pi$
$420$	0	0
$421$	−15.8360	−0.771798	−0.385899	−	0.922541i	$-0.626109\pi$
$421$	−15.8360	−0.771798	−0.385899	+	0.922541i	$0.626109\pi$
$422$	−67.9544	−3.30797
$423$	0	0
$424$	−18.9819	−0.921844
$425$	2.05475	0.0996699
$426$	0	0
$427$	−29.5345	−1.42927
$428$	−34.9518	−1.68946
$429$	0	0
$430$	−49.3672	−2.38070
$431$	−19.9233	−0.959674	−0.479837	−	0.877358i	$-0.659304\pi$
$431$	−19.9233	−0.959674	−0.479837	+	0.877358i	$0.659304\pi$
$432$	0	0
$433$	5.88331	0.282734	0.141367	−	0.989957i	$-0.454850\pi$
$433$	5.88331	0.282734	0.141367	+	0.989957i	$0.454850\pi$
$434$	−53.0733	−2.54760
$435$	0	0
$436$	−1.17658	−0.0563480
$437$	5.55926	0.265936
$438$	0	0
$439$	−22.6432	−1.08070	−0.540351	−	0.841440i	$-0.681708\pi$
$439$	−22.6432	−1.08070	−0.540351	+	0.841440i	$0.681708\pi$
$440$	31.9651	1.52388
$441$	0	0
$442$	1.39370	0.0662918
$443$	19.5044	0.926684	0.463342	−	0.886180i	$-0.346650\pi$
$443$	19.5044	0.926684	0.463342	+	0.886180i	$0.346650\pi$
$444$	0	0
$445$	67.2081	3.18597
$446$	5.12812	0.242824
$447$	0	0
$448$	38.4007	1.81426
$449$	−15.3358	−0.723741	−0.361870	−	0.932228i	$-0.617862\pi$
$449$	−15.3358	−0.723741	−0.361870	+	0.932228i	$0.617862\pi$
$450$	0	0
$451$	−10.5764	−0.498021
$452$	8.71759	0.410041
$453$	0	0
$454$	51.4904	2.41656
$455$	−50.0136	−2.34467
$456$	0	0
$457$	−1.69351	−0.0792192	−0.0396096	−	0.999215i	$-0.512611\pi$
$457$	−1.69351	−0.0792192	−0.0396096	+	0.999215i	$0.512611\pi$
$458$	−42.0945	−1.96695
$459$	0	0
$460$	−77.7861	−3.62680
$461$	23.4627	1.09277	0.546384	−	0.837535i	$-0.316004\pi$
$461$	23.4627	1.09277	0.546384	+	0.837535i	$0.316004\pi$
$462$	0	0
$463$	17.1364	0.796395	0.398197	−	0.917300i	$-0.369636\pi$
$463$	17.1364	0.796395	0.398197	+	0.917300i	$0.369636\pi$
$464$	17.4228	0.808831
$465$	0	0
$466$	−30.5702	−1.41614
$467$	24.1942	1.11958	0.559788	−	0.828636i	$-0.310883\pi$
$467$	24.1942	1.11958	0.559788	+	0.828636i	$0.310883\pi$
$468$	0	0
$469$	32.4771	1.49965
$470$	−49.9787	−2.30534
$471$	0	0
$472$	−17.6374	−0.811827
$473$	−9.36498	−0.430602
$474$	0	0
$475$	−13.0287	−0.597796
$476$	−2.34734	−0.107590
$477$	0	0
$478$	2.39589	0.109586
$479$	5.10566	0.233284	0.116642	−	0.993174i	$-0.462787\pi$
$479$	5.10566	0.233284	0.116642	+	0.993174i	$0.462787\pi$
$480$	0	0
$481$	11.5832	0.528150
$482$	−30.0005	−1.36648
$483$	0	0
$484$	−28.1109	−1.27777
$485$	−72.2169	−3.27920
$486$	0	0
$487$	4.69712	0.212847	0.106423	−	0.994321i	$-0.466060\pi$
$487$	4.69712	0.212847	0.106423	+	0.994321i	$0.466060\pi$
$488$	33.9748	1.53797
$489$	0	0
$490$	60.3976	2.72849
$491$	−32.8000	−1.48024	−0.740122	−	0.672473i	$-0.765232\pi$
$491$	−32.8000	−1.48024	−0.740122	+	0.672473i	$0.765232\pi$
$492$	0	0
$493$	1.20219	0.0541441
$494$	−8.83715	−0.397602
$495$	0	0
$496$	15.3352	0.688572
$497$	−50.3959	−2.26056
$498$	0	0
$499$	−18.9668	−0.849071	−0.424536	−	0.905411i	$-0.639563\pi$
$499$	−18.9668	−0.849071	−0.424536	+	0.905411i	$0.639563\pi$
$500$	105.492	4.71776
$501$	0	0
$502$	33.6209	1.50058
$503$	−15.7855	−0.703840	−0.351920	−	0.936030i	$-0.614471\pi$
$503$	−15.7855	−0.703840	−0.351920	+	0.936030i	$0.614471\pi$
$504$	0	0
$505$	−37.0818	−1.65012
$506$	−22.6464	−1.00675
$507$	0	0
$508$	−2.95757	−0.131221
$509$	−9.35294	−0.414562	−0.207281	−	0.978281i	$-0.566461\pi$
$509$	−9.35294	−0.414562	−0.207281	+	0.978281i	$0.566461\pi$
$510$	0	0
$511$	22.9810	1.01662
$512$	−26.7648	−1.18285
$513$	0	0
$514$	13.6050	0.600092
$515$	9.39323	0.413915
$516$	0	0
$517$	−9.48098	−0.416973
$518$	−29.9408	−1.31552
$519$	0	0
$520$	57.5328	2.52298
$521$	11.6796	0.511691	0.255845	−	0.966718i	$-0.417646\pi$
$521$	11.6796	0.511691	0.255845	+	0.966718i	$0.417646\pi$
$522$	0	0
$523$	−25.0355	−1.09473	−0.547363	−	0.836895i	$-0.684368\pi$
$523$	−25.0355	−1.09473	−0.547363	+	0.836895i	$0.684368\pi$
$524$	−61.8037	−2.69991
$525$	0	0
$526$	−12.3641	−0.539100
$527$	1.05815	0.0460938
$528$	0	0
$529$	2.64150	0.114848
$530$	−44.7960	−1.94581
$531$	0	0
$532$	14.8839	0.645300
$533$	−19.0360	−0.824541
$534$	0	0
$535$	−38.3784	−1.65924
$536$	−37.3598	−1.61370
$537$	0	0
$538$	4.05454	0.174804
$539$	11.4575	0.493508
$540$	0	0
$541$	22.3624	0.961433	0.480717	−	0.876876i	$-0.340377\pi$
$541$	22.3624	0.961433	0.480717	+	0.876876i	$0.340377\pi$
$542$	−28.0903	−1.20658
$543$	0	0
$544$	−0.402943	−0.0172761
$545$	−1.29193	−0.0553402
$546$	0	0
$547$	16.6583	0.712256	0.356128	−	0.934437i	$-0.384097\pi$
$547$	16.6583	0.712256	0.356128	+	0.934437i	$0.384097\pi$
$548$	−59.0047	−2.52056
$549$	0	0
$550$	53.0740	2.26308
$551$	−7.62282	−0.324743
$552$	0	0
$553$	−5.01732	−0.213358
$554$	60.4817	2.56962
$555$	0	0
$556$	55.7672	2.36506
$557$	41.6534	1.76491	0.882456	−	0.470394i	$-0.155888\pi$
$557$	41.6534	1.76491	0.882456	+	0.470394i	$0.155888\pi$
$558$	0	0
$559$	−16.8557	−0.712920
$560$	−37.3538	−1.57849
$561$	0	0
$562$	56.8387	2.39760
$563$	42.9853	1.81161	0.905807	−	0.423691i	$-0.139266\pi$
$563$	42.9853	1.81161	0.905807	+	0.423691i	$0.139266\pi$
$564$	0	0
$565$	9.57223	0.402707
$566$	68.8162	2.89256
$567$	0	0
$568$	57.9726	2.43247
$569$	26.8803	1.12688	0.563440	−	0.826157i	$-0.309477\pi$
$569$	26.8803	1.12688	0.563440	+	0.826157i	$0.309477\pi$
$570$	0	0
$571$	−21.5622	−0.902351	−0.451176	−	0.892435i	$-0.648995\pi$
$571$	−21.5622	−0.902351	−0.451176	+	0.892435i	$0.648995\pi$
$572$	23.4566	0.980771
$573$	0	0
$574$	49.2049	2.05377
$575$	−60.0933	−2.50607
$576$	0	0
$577$	42.1498	1.75472	0.877359	−	0.479835i	$-0.159304\pi$
$577$	42.1498	1.75472	0.877359	+	0.479835i	$0.159304\pi$
$578$	−40.6584	−1.69116
$579$	0	0
$580$	106.660	4.42881
$581$	54.1486	2.24646
$582$	0	0
$583$	−8.49782	−0.351944
$584$	−26.4360	−1.09393
$585$	0	0
$586$	−13.5690	−0.560530
$587$	18.8112	0.776420	0.388210	−	0.921571i	$-0.373094\pi$
$587$	18.8112	0.776420	0.388210	+	0.921571i	$0.373094\pi$
$588$	0	0
$589$	−6.70949	−0.276460
$590$	−41.6230	−1.71359
$591$	0	0
$592$	8.65121	0.355563
$593$	17.1975	0.706218	0.353109	−	0.935582i	$-0.385125\pi$
$593$	17.1975	0.706218	0.353109	+	0.935582i	$0.385125\pi$
$594$	0	0
$595$	−2.57746	−0.105666
$596$	61.9717	2.53846
$597$	0	0
$598$	−40.7604	−1.66682
$599$	−7.02859	−0.287181	−0.143590	−	0.989637i	$-0.545865\pi$
$599$	−7.02859	−0.287181	−0.143590	+	0.989637i	$0.545865\pi$
$600$	0	0
$601$	−32.4567	−1.32394	−0.661968	−	0.749532i	$-0.730279\pi$
$601$	−32.4567	−1.32394	−0.661968	+	0.749532i	$0.730279\pi$
$602$	43.5692	1.77575
$603$	0	0
$604$	−42.9546	−1.74780
$605$	−30.8668	−1.25491
$606$	0	0
$607$	11.6545	0.473040	0.236520	−	0.971627i	$-0.423993\pi$
$607$	11.6545	0.473040	0.236520	+	0.971627i	$0.423993\pi$
$608$	2.55497	0.103618
$609$	0	0
$610$	80.1781	3.24632
$611$	−17.0645	−0.690355
$612$	0	0
$613$	−2.82297	−0.114019	−0.0570093	−	0.998374i	$-0.518156\pi$
$613$	−2.82297	−0.114019	−0.0570093	+	0.998374i	$0.518156\pi$
$614$	−42.5698	−1.71798
$615$	0	0
$616$	−28.2109	−1.13665
$617$	23.5462	0.947936	0.473968	−	0.880542i	$-0.342821\pi$
$617$	23.5462	0.947936	0.473968	+	0.880542i	$0.342821\pi$
$618$	0	0
$619$	38.2730	1.53832	0.769162	−	0.639054i	$-0.220674\pi$
$619$	38.2730	1.53832	0.769162	+	0.639054i	$0.220674\pi$
$620$	93.8803	3.77032
$621$	0	0
$622$	59.5217	2.38660
$623$	−59.3147	−2.37640
$624$	0	0
$625$	56.4977	2.25991
$626$	−70.3224	−2.81065
$627$	0	0
$628$	−39.4854	−1.57564
$629$	0.596945	0.0238018
$630$	0	0
$631$	14.2574	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$631$	14.2574	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$632$	5.77164	0.229584
$633$	0	0
$634$	−54.5928	−2.16816
$635$	−3.24752	−0.128874
$636$	0	0
$637$	20.6219	0.817068
$638$	31.0526	1.22938
$639$	0	0
$640$	−85.1314	−3.36512
$641$	−31.3172	−1.23696	−0.618478	−	0.785802i	$-0.712250\pi$
$641$	−31.3172	−1.23696	−0.618478	+	0.785802i	$0.712250\pi$
$642$	0	0
$643$	50.4477	1.98946	0.994732	−	0.102512i	$-0.0326880\pi$
$643$	50.4477	1.98946	0.994732	+	0.102512i	$0.0326880\pi$
$644$	68.6505	2.70521
$645$	0	0
$646$	−0.455425	−0.0179185
$647$	2.82116	0.110911	0.0554555	−	0.998461i	$-0.482339\pi$
$647$	2.82116	0.110911	0.0554555	+	0.998461i	$0.482339\pi$
$648$	0	0
$649$	−7.89590	−0.309941
$650$	95.5260	3.74684
$651$	0	0
$652$	71.1344	2.78584
$653$	−43.4096	−1.69875	−0.849374	−	0.527792i	$-0.823020\pi$
$653$	−43.4096	−1.69875	−0.849374	+	0.527792i	$0.823020\pi$
$654$	0	0
$655$	−67.8627	−2.65162
$656$	−14.2175	−0.555099
$657$	0	0
$658$	44.1089	1.71954
$659$	36.4540	1.42005	0.710024	−	0.704178i	$-0.248684\pi$
$659$	36.4540	1.42005	0.710024	+	0.704178i	$0.248684\pi$
$660$	0	0
$661$	13.0915	0.509201	0.254600	−	0.967046i	$-0.418056\pi$
$661$	13.0915	0.509201	0.254600	+	0.967046i	$0.418056\pi$
$662$	55.7589	2.16713
$663$	0	0
$664$	−62.2895	−2.41730
$665$	16.3431	0.633758
$666$	0	0
$667$	−35.1594	−1.36138
$668$	5.75366	0.222616
$669$	0	0
$670$	−88.1663	−3.40616
$671$	15.2098	0.587169
$672$	0	0
$673$	3.94420	0.152038	0.0760188	−	0.997106i	$-0.475779\pi$
$673$	3.94420	0.152038	0.0760188	+	0.997106i	$0.475779\pi$
$674$	77.7913	2.99641
$675$	0	0
$676$	−6.40517	−0.246353
$677$	34.7858	1.33693	0.668464	−	0.743744i	$-0.266952\pi$
$677$	34.7858	1.33693	0.668464	+	0.743744i	$0.266952\pi$
$678$	0	0
$679$	63.7353	2.44594
$680$	2.96497	0.113701
$681$	0	0
$682$	27.3320	1.04660
$683$	−32.7697	−1.25390	−0.626949	−	0.779060i	$-0.715697\pi$
$683$	−32.7697	−1.25390	−0.626949	+	0.779060i	$0.715697\pi$
$684$	0	0
$685$	−64.7893	−2.47547
$686$	7.48561	0.285802
$687$	0	0
$688$	−12.5891	−0.479954
$689$	−15.2949	−0.582690
$690$	0	0
$691$	−27.6403	−1.05149	−0.525744	−	0.850643i	$-0.676213\pi$
$691$	−27.6403	−1.05149	−0.525744	+	0.850643i	$0.676213\pi$
$692$	−59.9847	−2.28027
$693$	0	0
$694$	−20.6522	−0.783948
$695$	61.2345	2.32276
$696$	0	0
$697$	−0.981025	−0.0371590
$698$	−49.2712	−1.86494
$699$	0	0
$700$	−160.889	−6.08103
$701$	−17.8186	−0.673000	−0.336500	−	0.941683i	$-0.609243\pi$
$701$	−17.8186	−0.673000	−0.336500	+	0.941683i	$0.609243\pi$
$702$	0	0
$703$	−3.78509	−0.142757
$704$	−19.7758	−0.745327
$705$	0	0
$706$	67.4506	2.53854
$707$	32.7267	1.23081
$708$	0	0
$709$	28.5417	1.07190	0.535952	−	0.844248i	$-0.319953\pi$
$709$	28.5417	1.07190	0.535952	+	0.844248i	$0.319953\pi$
$710$	136.811	5.13443
$711$	0	0
$712$	68.2323	2.55711
$713$	−30.9468	−1.15897
$714$	0	0
$715$	25.7563	0.963229
$716$	33.3793	1.24744
$717$	0	0
$718$	−32.4913	−1.21256
$719$	16.9208	0.631040	0.315520	−	0.948919i	$-0.397821\pi$
$719$	16.9208	0.631040	0.315520	+	0.948919i	$0.397821\pi$
$720$	0	0
$721$	−8.29004	−0.308737
$722$	−42.6342	−1.58668
$723$	0	0
$724$	39.3570	1.46269
$725$	82.3996	3.06024
$726$	0	0
$727$	−3.88604	−0.144125	−0.0720627	−	0.997400i	$-0.522958\pi$
$727$	−3.88604	−0.144125	−0.0720627	+	0.997400i	$0.522958\pi$
$728$	−50.7758	−1.88188
$729$	0	0
$730$	−62.3872	−2.30905
$731$	−0.868663	−0.0321287
$732$	0	0
$733$	11.3841	0.420480	0.210240	−	0.977650i	$-0.432575\pi$
$733$	11.3841	0.420480	0.210240	+	0.977650i	$0.432575\pi$
$734$	−61.8109	−2.28148
$735$	0	0
$736$	11.7845	0.434383
$737$	−16.7252	−0.616081
$738$	0	0
$739$	31.2138	1.14822	0.574109	−	0.818779i	$-0.305349\pi$
$739$	31.2138	1.14822	0.574109	+	0.818779i	$0.305349\pi$
$740$	52.9616	1.94691
$741$	0	0
$742$	39.5349	1.45137
$743$	−29.8800	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$743$	−29.8800	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$744$	0	0
$745$	68.0472	2.49306
$746$	85.2368	3.12074
$747$	0	0
$748$	1.20884	0.0441997
$749$	33.8710	1.23762
$750$	0	0
$751$	0.548324	0.0200086	0.0100043	−	0.999950i	$-0.496815\pi$
$751$	0.548324	0.0200086	0.0100043	+	0.999950i	$0.496815\pi$
$752$	−12.7450	−0.464763
$753$	0	0
$754$	55.8904	2.03541
$755$	−47.1657	−1.71654
$756$	0	0
$757$	−38.9526	−1.41576	−0.707878	−	0.706335i	$-0.750347\pi$
$757$	−38.9526	−1.41576	−0.707878	+	0.706335i	$0.750347\pi$
$758$	−6.05868	−0.220061
$759$	0	0
$760$	−18.8002	−0.681954
$761$	6.47569	0.234743	0.117372	−	0.993088i	$-0.462553\pi$
$761$	6.47569	0.234743	0.117372	+	0.993088i	$0.462553\pi$
$762$	0	0
$763$	1.14020	0.0412779
$764$	79.2477	2.86708
$765$	0	0
$766$	−32.9690	−1.19122
$767$	−14.2116	−0.513149
$768$	0	0
$769$	9.89337	0.356764	0.178382	−	0.983961i	$-0.442914\pi$
$769$	9.89337	0.356764	0.178382	+	0.983961i	$0.442914\pi$
$770$	−66.5757	−2.39922
$771$	0	0
$772$	−20.7611	−0.747210
$773$	12.2179	0.439447	0.219723	−	0.975562i	$-0.429485\pi$
$773$	12.2179	0.439447	0.219723	+	0.975562i	$0.429485\pi$
$774$	0	0
$775$	72.5268	2.60524
$776$	−73.3175	−2.63195
$777$	0	0
$778$	−55.2722	−1.98160
$779$	6.22045	0.222871
$780$	0	0
$781$	25.9531	0.928676
$782$	−2.10060	−0.0751173
$783$	0	0
$784$	15.4019	0.550069
$785$	−43.3565	−1.54746
$786$	0	0
$787$	−32.8255	−1.17010	−0.585052	−	0.810996i	$-0.698926\pi$
$787$	−32.8255	−1.17010	−0.585052	+	0.810996i	$0.698926\pi$
$788$	61.7122	2.19841
$789$	0	0
$790$	13.6207	0.484601
$791$	−8.44801	−0.300377
$792$	0	0
$793$	27.3756	0.972137
$794$	38.7258	1.37433
$795$	0	0
$796$	77.1203	2.73346
$797$	52.9963	1.87723	0.938613	−	0.344972i	$-0.112111\pi$
$797$	52.9963	1.87723	0.938613	+	0.344972i	$0.112111\pi$
$798$	0	0
$799$	−0.879423	−0.0311117
$800$	−27.6182	−0.976449
$801$	0	0
$802$	−33.3068	−1.17611
$803$	−11.8349	−0.417644
$804$	0	0
$805$	75.3808	2.65682
$806$	49.1939	1.73278
$807$	0	0
$808$	−37.6469	−1.32441
$809$	8.64210	0.303840	0.151920	−	0.988393i	$-0.451454\pi$
$809$	8.64210	0.303840	0.151920	+	0.988393i	$0.451454\pi$
$810$	0	0
$811$	15.2237	0.534575	0.267287	−	0.963617i	$-0.413873\pi$
$811$	15.2237	0.534575	0.267287	+	0.963617i	$0.413873\pi$
$812$	−94.1331	−3.30342
$813$	0	0
$814$	15.4191	0.540438
$815$	78.1082	2.73601
$816$	0	0
$817$	5.50798	0.192700
$818$	−18.6812	−0.653172
$819$	0	0
$820$	−87.0376	−3.03948
$821$	−12.0526	−0.420637	−0.210318	−	0.977633i	$-0.567450\pi$
$821$	−12.0526	−0.420637	−0.210318	+	0.977633i	$0.567450\pi$
$822$	0	0
$823$	−48.1644	−1.67891	−0.839453	−	0.543432i	$-0.817124\pi$
$823$	−48.1644	−1.67891	−0.839453	+	0.543432i	$0.817124\pi$
$824$	9.53639	0.332216
$825$	0	0
$826$	36.7345	1.27816
$827$	−1.62044	−0.0563481	−0.0281740	−	0.999603i	$-0.508969\pi$
$827$	−1.62044	−0.0563481	−0.0281740	+	0.999603i	$0.508969\pi$
$828$	0	0
$829$	29.6369	1.02933	0.514666	−	0.857391i	$-0.327916\pi$
$829$	29.6369	1.02933	0.514666	+	0.857391i	$0.327916\pi$
$830$	−146.999	−5.10240
$831$	0	0
$832$	−35.5937	−1.23399
$833$	1.06275	0.0368222
$834$	0	0
$835$	6.31773	0.218634
$836$	−7.66499	−0.265099
$837$	0	0
$838$	10.6538	0.368031
$839$	39.2976	1.35671	0.678353	−	0.734737i	$-0.262694\pi$
$839$	39.2976	1.35671	0.678353	+	0.734737i	$0.262694\pi$
$840$	0	0
$841$	19.2104	0.662428
$842$	−37.9413	−1.30754
$843$	0	0
$844$	−106.086	−3.65162
$845$	−7.03311	−0.241946
$846$	0	0
$847$	27.2416	0.936033
$848$	−11.4234	−0.392280
$849$	0	0
$850$	4.92296	0.168856
$851$	−17.4583	−0.598463
$852$	0	0
$853$	−3.53146	−0.120915	−0.0604574	−	0.998171i	$-0.519256\pi$
$853$	−3.53146	−0.120915	−0.0604574	+	0.998171i	$0.519256\pi$
$854$	−70.7615	−2.42141
$855$	0	0
$856$	−38.9633	−1.33174
$857$	22.3714	0.764191	0.382096	−	0.924123i	$-0.375203\pi$
$857$	22.3714	0.764191	0.382096	+	0.924123i	$0.375203\pi$
$858$	0	0
$859$	6.24869	0.213203	0.106601	−	0.994302i	$-0.466003\pi$
$859$	6.24869	0.213203	0.106601	+	0.994302i	$0.466003\pi$
$860$	−77.0687	−2.62802
$861$	0	0
$862$	−47.7342	−1.62583
$863$	23.4800	0.799270	0.399635	−	0.916674i	$-0.369137\pi$
$863$	23.4800	0.799270	0.399635	+	0.916674i	$0.369137\pi$
$864$	0	0
$865$	−65.8653	−2.23949
$866$	14.0958	0.478994
$867$	0	0
$868$	−82.8545	−2.81226
$869$	2.58384	0.0876509
$870$	0	0
$871$	−30.1031	−1.02000
$872$	−1.31162	−0.0444170
$873$	0	0
$874$	13.3194	0.450535
$875$	−102.230	−3.45601
$876$	0	0
$877$	−22.1035	−0.746382	−0.373191	−	0.927755i	$-0.621736\pi$
$877$	−22.1035	−0.746382	−0.373191	+	0.927755i	$0.621736\pi$
$878$	−54.2507	−1.83087
$879$	0	0
$880$	19.2367	0.648468
$881$	18.1078	0.610068	0.305034	−	0.952341i	$-0.401332\pi$
$881$	18.1078	0.610068	0.305034	+	0.952341i	$0.401332\pi$
$882$	0	0
$883$	1.96256	0.0660455	0.0330227	−	0.999455i	$-0.489487\pi$
$883$	1.96256	0.0660455	0.0330227	+	0.999455i	$0.489487\pi$
$884$	2.17576	0.0731786
$885$	0	0
$886$	46.7306	1.56994
$887$	−48.6192	−1.63247	−0.816236	−	0.577718i	$-0.803943\pi$
$887$	−48.6192	−1.63247	−0.816236	+	0.577718i	$0.803943\pi$
$888$	0	0
$889$	2.86611	0.0961262
$890$	161.023	5.39751
$891$	0	0
$892$	8.00568	0.268050
$893$	5.57621	0.186601
$894$	0	0
$895$	36.6517	1.22513
$896$	75.1331	2.51002
$897$	0	0
$898$	−36.7429	−1.22613
$899$	42.4340	1.41525
$900$	0	0
$901$	−0.788228	−0.0262597
$902$	−25.3398	−0.843723
$903$	0	0
$904$	9.71812	0.323220
$905$	43.2155	1.43653
$906$	0	0
$907$	17.3975	0.577675	0.288837	−	0.957378i	$-0.406731\pi$
$907$	17.3975	0.577675	0.288837	+	0.957378i	$0.406731\pi$
$908$	80.3832	2.66761
$909$	0	0
$910$	−119.827	−3.97223
$911$	9.19217	0.304550	0.152275	−	0.988338i	$-0.451340\pi$
$911$	9.19217	0.304550	0.152275	+	0.988338i	$0.451340\pi$
$912$	0	0
$913$	−27.8857	−0.922883
$914$	−4.05748	−0.134209
$915$	0	0
$916$	−65.7150	−2.17129
$917$	59.8925	1.97783
$918$	0	0
$919$	47.2057	1.55717	0.778587	−	0.627537i	$-0.215937\pi$
$919$	47.2057	1.55717	0.778587	+	0.627537i	$0.215937\pi$
$920$	−86.7137	−2.85887
$921$	0	0
$922$	56.2142	1.85132
$923$	46.7121	1.53755
$924$	0	0
$925$	40.9153	1.34529
$926$	41.0569	1.34921
$927$	0	0
$928$	−16.1588	−0.530440
$929$	15.7119	0.515490	0.257745	−	0.966213i	$-0.417021\pi$
$929$	15.7119	0.515490	0.257745	+	0.966213i	$0.417021\pi$
$930$	0	0
$931$	−6.73867	−0.220851
$932$	−47.7242	−1.56326
$933$	0	0
$934$	57.9668	1.89673
$935$	1.32736	0.0434092
$936$	0	0
$937$	−39.2075	−1.28085	−0.640427	−	0.768019i	$-0.721242\pi$
$937$	−39.2075	−1.28085	−0.640427	+	0.768019i	$0.721242\pi$
$938$	77.8116	2.54064
$939$	0	0
$940$	−78.0233	−2.54484
$941$	13.4619	0.438844	0.219422	−	0.975630i	$-0.429583\pi$
$941$	13.4619	0.438844	0.219422	+	0.975630i	$0.429583\pi$
$942$	0	0
$943$	28.6912	0.934312
$944$	−10.6142	−0.345464
$945$	0	0
$946$	−22.4375	−0.729506
$947$	−9.05254	−0.294168	−0.147084	−	0.989124i	$-0.546989\pi$
$947$	−9.05254	−0.294168	−0.147084	+	0.989124i	$0.546989\pi$
$948$	0	0
$949$	−21.3012	−0.691465
$950$	−31.2153	−1.01276
$951$	0	0
$952$	−2.61675	−0.0848092
$953$	−29.5658	−0.957729	−0.478864	−	0.877889i	$-0.658951\pi$
$953$	−29.5658	−0.957729	−0.478864	+	0.877889i	$0.658951\pi$
$954$	0	0
$955$	87.0169	2.81580
$956$	3.74030	0.120970
$957$	0	0
$958$	12.2326	0.395218
$959$	57.1801	1.84644
$960$	0	0
$961$	6.34978	0.204831
$962$	27.7522	0.894767
$963$	0	0
$964$	−46.8347	−1.50844
$965$	−22.7965	−0.733845
$966$	0	0
$967$	20.9682	0.674290	0.337145	−	0.941453i	$-0.390539\pi$
$967$	20.9682	0.674290	0.337145	+	0.941453i	$0.390539\pi$
$968$	−31.3372	−1.00722
$969$	0	0
$970$	−173.024	−5.55547
$971$	14.5417	0.466664	0.233332	−	0.972397i	$-0.425037\pi$
$971$	14.5417	0.466664	0.233332	+	0.972397i	$0.425037\pi$
$972$	0	0
$973$	−54.0427	−1.73253
$974$	11.2538	0.360595
$975$	0	0
$976$	20.4461	0.654464
$977$	42.9015	1.37254	0.686270	−	0.727347i	$-0.259247\pi$
$977$	42.9015	1.37254	0.686270	+	0.727347i	$0.259247\pi$
$978$	0	0
$979$	30.5462	0.976261
$980$	94.2886	3.01194
$981$	0	0
$982$	−78.5853	−2.50776
$983$	8.10974	0.258661	0.129330	−	0.991602i	$-0.458717\pi$
$983$	8.10974	0.258661	0.129330	+	0.991602i	$0.458717\pi$
$984$	0	0
$985$	67.7623	2.15909
$986$	2.88033	0.0917283
$987$	0	0
$988$	−13.7960	−0.438908
$989$	25.4050	0.807831
$990$	0	0
$991$	39.5827	1.25738	0.628692	−	0.777654i	$-0.283591\pi$
$991$	39.5827	1.25738	0.628692	+	0.777654i	$0.283591\pi$
$992$	−14.2228	−0.451573
$993$	0	0
$994$	−120.743	−3.82974
$995$	84.6809	2.68456
$996$	0	0
$997$	35.5573	1.12611	0.563056	−	0.826419i	$-0.309626\pi$
$997$	35.5573	1.12611	0.563056	+	0.826419i	$0.309626\pi$
$998$	−45.4425	−1.43846
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.17	yes	20
3.2	odd	2		2151.2.a.j.1.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.4	✓	20	3.2	odd	2
2151.2.a.k.1.17	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39589 q^{2} +3.74030 q^{4} +4.10699 q^{5} -3.62464 q^{7} +4.16958 q^{8} +O(q^{10})\)	\(q+2.39589 q^{2} +3.74030 q^{4} +4.10699 q^{5} -3.62464 q^{7} +4.16958 q^{8} +9.83991 q^{10} +1.86664 q^{11} +3.35969 q^{13} -8.68425 q^{14} +2.50926 q^{16} +0.173143 q^{17} -1.09786 q^{19} +15.3614 q^{20} +4.47226 q^{22} -5.06374 q^{23} +11.8674 q^{25} +8.04946 q^{26} -13.5573 q^{28} +6.94337 q^{29} +6.11145 q^{31} -2.32723 q^{32} +0.414831 q^{34} -14.8864 q^{35} +3.44771 q^{37} -2.63035 q^{38} +17.1244 q^{40} -5.66600 q^{41} -5.01704 q^{43} +6.98178 q^{44} -12.1322 q^{46} -5.07918 q^{47} +6.13803 q^{49} +28.4330 q^{50} +12.5663 q^{52} -4.55248 q^{53} +7.66626 q^{55} -15.1132 q^{56} +16.6356 q^{58} -4.23002 q^{59} +8.14825 q^{61} +14.6424 q^{62} -10.5943 q^{64} +13.7982 q^{65} -8.96008 q^{67} +0.647606 q^{68} -35.6661 q^{70} +13.9037 q^{71} -6.34022 q^{73} +8.26034 q^{74} -4.10632 q^{76} -6.76589 q^{77} +1.38423 q^{79} +10.3055 q^{80} -13.5751 q^{82} -14.9390 q^{83} +0.711095 q^{85} -12.0203 q^{86} +7.78309 q^{88} +16.3643 q^{89} -12.1777 q^{91} -18.9399 q^{92} -12.1692 q^{94} -4.50888 q^{95} -17.5839 q^{97} +14.7061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more